Spectral resolution of the linear test - The KLS theorem and its methods

Part of the moment-map mechanism, Chapter The moment map: CMH and the linear test; the reading order is on the full proofs page.

Overview. This dossier proves Lemma 16.1 in the refined form Theorem D10.1. On a single regular isotropic compact-target moment map, it resolves the gate matrices $\mathsf N,\mathsf D,\mathsf R$ along the spectrum of the Stein operator. The key input is that the Hessian columns $Ha$ solve a weak eigen-type column equation, obtained by differentiating the Monge–Ampère equation twice. The dossier also proves the dimension-dependent retention $\mathsf R\succeq\mathsf N-n\Id$ and the sharp bound $\mathsf R\succeq\mathsf D$ on finite products. It relies on the published imports Theorem 4.1 and the Hessian bound (D10.5). It proves no universal bootstrap, no bound on $\CMH$ , and no strong-form column equation on the general class.

Pointwise calculus: from (D10.20), Lemma D10.1 and Lemma D10.2 identify the source generator with the Stein generator. Lemma D10.3 gives $\calL H+H=A+Q$ with $A,Q\succeq0$ .
Cutoff calculus (Lemma D10.6–Lemma D10.10): finiteness of $\mathsf D$ , the trace identity (D10.30) and the normalization $\int(A+Q)\,\dd\eta=\Id$ . The same lemmas give the weak column equation (D10.13).
Lemma D10.11 and Lemma D10.12: the coordinates $u_b$ are eigenfunctions with eigenvalue 1, and Brascamp–Lieb makes 1 the exact gap.
Using Steps 2–3 together with Lemma D10.13 and Lemma D10.14, the orthogonal decomposition of $Ha$ gives the resolution (D10.15). Proposition D10.1 identifies $\mathsf R$ as an anticommutator integral.
Corollaries C1–C5 follow from Step 4. They include $\mathsf R\preceq\Id$ and the channel criterion (D10.16) for $(\mathrm{AB})_{\rho,\beta}$ .
Part (f): the cyclic square Lemma D10.15, used under the trace only, with Steps 2 and 4 gives $\Tr\mathsf R\ge\Tr\mathsf D$ , $\mathsf R\succeq\mathsf N-n\Id$ and $Q_{\mathrm{lin}}\le n+1$ .
Part (g): on products everything is diagonal, and $\mathsf R-\mathsf D$ reduces to a nonnegative curvature integral.

Scope. This dossier proves the candidate Lemma 16.1 of Chapter The moment map: CMH and the linear test: on one fixed regular isotropic compact-target moment map, the three linear-sector gate matrices resolve exactly along the spectrum of the Stein operator, with the columns of the moment Hessian solving, in the weak (closed-form) sense stated in the manuscript lemma, an eigen-type equation forced by the differentiated Monge–Ampère structure. It also proves the unconditional dimension-dependent retention $\mathsf R\succeq\mathsf N-n\Id$ (hence $Q_{\mathrm{lin}}\le n+1$ ) and the sharp retention $\mathsf R\succeq\mathsf D$ on regular finite products of one-dimensional laws.

What this dossier does not prove. It proves no anisotropic bootstrap $(\mathrm{AB})_{\rho,\beta}$ with universal (dimension-free) constants; no bound on $Q_{\mathrm{lin}}$ beyond $n+1$ ; no bound on $\CMH$ , no statement about Conjecture 0.1 (KLS), no statement about conj:gate-zero beyond the exact identities below, and nothing about the trace-upgrade cluster (conj:trace-upgrade, high-rank conj:stein-weighted, conj:product-alignment), which is not opened here. All statements are stationary identities and inequalities at one fixed map of the regular class. No numerical evidence is used anywhere.

0. Standing setting, imports, and notation¶

Definition D10.1 (Regular isotropic compact-target moment map)

Let $P\subset\R^n$ be a convex body and let

\mu(\dd x)=e^{-V(x)}\one_{\operatorname{int}P}(x)\,\dd x

(D10.1)

be a probability measure with $V\in C^\infty(\R^n)$ , such that $\mu$ is centered ( $\E_\mu X=0$ ), isotropic ( $\Cov\mu=\Id$ ), and log-concave on its support ( $D^2V\succeq0$ on $\operatorname{int}P$ ). By the imported compact-target regularity theorem (Theorem 4.1; Berman & Berndtsson, 2013Fathi, 2019) the canonical moment potential $\psi$ is smooth and strictly convex on $\R^n$ ,

\nabla\psi:\R^n\longrightarrow\operatorname{int}P

(D10.2)

is a diffeomorphism, and the target-coordinate Hessian

\tau(x)=D^2\psi\bigl((\nabla\psi)^{-1}(x)\bigr)

(D10.3)

is a smooth positive symmetric Stein kernel: $\Div_\mu\tau=-x$ distributionally with weak zero boundary flux, and $\E_\mu\tau=\Cov\mu=\Id$ . We write

\eta(\dd y)=e^{-\psi(y)}\,\dd y,\qquad H(y)=D^2\psi(y)\succ0,\qquad (\nabla\psi)_\#\eta=\mu ,

(D10.4)

so $\eta$ is a probability measure (the moment-measure normalization) and $\tau(\nabla\psi(y))=H(y)$ . Set $R=R(P)=\max_{x\in P}\abs x$ , $c_V=\sup_{\operatorname{int}P}\abs{\nabla V}<\infty$ and $c_{V''}=\sup_{\operatorname{int}P}\norm{D^2V}_\op<\infty$ (finite because $V$ is smooth on $\R^n$ and $\overline P$ is compact).

Remark D10.1 (Imported pointwise Hessian bound)

Theorem 1.1 of Klartag, 2014 gives $\Delta\psi\le2R(P)^2$ on $\R^n$ for a centered log-concave target on a bounded convex $P$ whose potential $V$ is smooth with all derivatives bounded on $P$ (his conditions (1), satisfied here because $V\in C^\infty(\R^n)$ and $\overline P$ is compact); since $H\succ0$ , this yields $H\preceq(\Tr H)\Id\preceq2R(P)^2\Id$ , i.e. the pointwise bound

0\prec H(y)\preceq 2R(P)^2\,\Id\qquad\text{for all }y\in\R^n.

(D10.5)

This published import is load-bearing for every integrability statement below; we flag each use. All constants produced by (D10.5) may depend on the map and on $n$ ; the lemma asserts no universality of constants.

Operator data. Following Definition 16.1 and the certified conventions of Chapter The moment map: CMH and the linear test, the Stein generator in target coordinates is $L_\mu g=\Div_\mu(\tau\nabla g)=\Tr(\tau D^2g)-x\cdot\nabla g$ , the form core is

\mathscr D=\bigl\{F|_{\operatorname{int}P}:F\in\R+C_c^\infty(\R^n)\bigr\}, \qquad \calE^0(f,g)=\E_\mu\inner{\tau\nabla f}{\nabla g},

(D10.6)

$\calE$ is the closed form generated by $(\calE^0,\mathscr D)$ (closable by the certified normalization dossier), and $\Aop\ge0$ is its self-adjoint operator on $L^2(\mu)$ , with $\ker\Aop=\R\one$ because $\tau\succ0$ and $\operatorname{int}P$ is connected. Every core element has finite energy since $\calE^0(f,f)\le\norm{\nabla f}_\infty^2\Tr\E_\mu\tau =\norm{\nabla f}_\infty^2\,n<\infty$ . The unitary map

U:L^2(\mu)\to L^2(\eta),\qquad Ug=g\circ\nabla\psi

(D10.7)

(isometric because $(\nabla\psi)_\#\eta=\mu$ , surjective because $\nabla\psi$ is a diffeomorphism) transports $\calE$ , $\Aop$ , and all spectral objects to $L^2(\eta)$ ; we use the same symbols on both sides and say in which coordinates a computation is performed. No operator inverse (and no pseudoinverse of $\Aop$ ) is used anywhere in this dossier.

The gate matrices. Write $\psi_{i}$ , $\psi_{ij}=H_{ij}$ , $\psi_{ijk}=\partial_kH_{ij}$ for source derivatives, Einstein summation throughout, and $(H^{ij})=H^{-1}$ (pointwise matrix inverse of the positive matrix $H(y)$ ; this is not an operator inverse). Define the source matrices

A=H\,(D^2V\circ\nabla\psi)\,H,\qquad Q_{k\ell}=\Tr\bigl(H^{-1}\partial_kH\,H^{-1}\partial_\ell H\bigr),

(D10.8)

and the constant symmetric matrices

\mathsf N=\int H^2\,\dd\eta,\qquad \mathsf D_{k\ell}=\int H^{bc}(\partial_bH)_{km}(\partial_cH)_{m\ell}\,\dd\eta,\qquad \mathsf R=\mathsf N-\mathsf D,

(D10.9)

following the manuscript statement, which defines $\mathsf R=\mathsf N-\mathsf D$ ; Proposition D10.1 identifies $\mathsf R$ with the anticommutator integral $\tfrac12\int\{H,A+Q\}\,\dd\eta$ used by the route file. $\mathsf N$ is finite by (D10.5); finiteness of $\mathsf D$ is proved below (Lemma D10.8), not assumed. Finally, in isotropic position the linear CMH quotient is

Q_{\mathrm{lin}}(\mu)=\lmax(\mathsf N) =\sup_{\abs a=1}\int\abs{Ha}^2\dd\eta .

(D10.10)

Refined statement.

Theorem D10.1 (Spectral resolution of the linear-sector gate matrices; refined form of Lemma 16.1)

Let the map be as in Definition D10.1, with the import (D10.5). For a unit vector $a\in\R^n$ put $u_a=\inner a{\nabla\psi}$ , so that the column is $Ha=\nabla\inner a{\nabla\psi}=\nabla u_a$ (componentwise $(Ha)_i=H_{ik}a_k$ ; for coordinate vectors, $u_b=\partial_b\psi$ ). Then:

(a) (Symmetric form and pullback.) Pointwise on $\R^n$ ,

\calL f:=H^{ij}f_{ij}-(V_i\circ\nabla\psi)f_i =e^{\psi}\,\partial_i\bigl(e^{-\psi}H^{ij}\partial_jf\bigr),

(D10.11)

and for every smooth $g$ on $\operatorname{int}P$ , $\nabla_y(g\circ\nabla\psi)=H\,(\nabla_xg\circ\nabla\psi)$ and $\calL(g\circ\nabla\psi)=(L_\mu g)\circ\nabla\psi$ : $\calL$ is the source pullback of the certified Stein generator, $U\Aop U^{-1}$ is the self-adjoint operator of the source form $\calE(f)=\int\inner{H^{-1}\nabla f}{\nabla f}\dd\eta$ , and the CMH numerator field $\tau\nabla_xg$ pulls back to the plain Euclidean source gradient $\nabla_yf$ of $f=g\circ\nabla\psi$ .

(b) (Eigenfunctions and gap.) The functions $u_1,\dots,u_n$ are bounded, centered, orthonormal in $L^2(\eta)$ , lie in $\Dom(\Aop)$ , and satisfy $\Aop u_b=u_b$ . Brascamp–Lieb gives $\inner f{\Aop f}\ge\norm f_2^2$ for every centered $f\in\Dom(\calE)$ ; hence the spectrum of $\Aop$ on $\one^\perp$ is contained in $[1,\infty)$ and the spectral gap of $\Aop$ equals 1 exactly, attained at each $u_b$ .

(c) (Weak column equation, normalization, orthogonality.) $A\succeq0$ and $Q\succeq0$ pointwise, $\calL H+H=A+Q$ pointwise, the entries of $A+Q$ are in $L^1(\eta)$ with

\int(A+Q)\,\dd\eta=\Id ,

(D10.12)

so in particular $(A+Q)a\in L^1(\eta;\R^n)$ . The columns $Ha=\nabla\inner a{\nabla\psi}$ lie in the form domain $\Dom(\calE)$ , componentwise, and satisfy the column equation in the weak form

\calE\bigl(f,(Ha)_i\bigr) =\bigl\langle f,(Ha)_i\bigr\rangle_{L^2(\eta)} -\bigl\langle f,\bigl((A+Q)a\bigr)_i\bigr\rangle_{L^2(\eta),L^1}

(D10.13)

for every bounded smooth finite-energy $f$ , i.e. every $f\in C^\infty(\R^n)\cap L^\infty$ with $\int\inner{H^{-1}\nabla f}{\nabla f}\dd\eta<\infty$ (the class $\mathfrak B$ of Lemma D10.10, every member of which lies in $\Dom(\calE)$ ; in particular every core element and every $u_b$ ). Consequently $(A+Q)a$ is orthogonal to every $u_b$ : $\int u_b\,\bigl((A+Q)a\bigr)_i\,\dd\eta=0$ for all $i,b$ . This weak form is exactly the column equation asserted by Lemma 16.1; no membership $(Ha)_i\in\Dom(\Aop)$ is asserted (Remark D10.2).

(d) (Resolution.) Decompose, componentwise and orthogonally in $L^2(\eta)$ ,

Ha=a\cdot\one+\sum_b(M_a)_{\cdot b}\,u_b+v, \qquad (M_a)_{ib}=\inner{(Ha)_i}{u_b}=a_k\int\psi_{kib}\,\dd\eta,

(D10.14)

with each $v_i\perp\one,u_1,\dots,u_n$ ; here $U:=\operatorname{span}\{u_1,\dots,u_n\}$ (which may be a proper subspace of the full eigenspace $\ker(\Aop-1)$ ; nothing below assumes otherwise), and the tensor $M_{kib}=\int\psi_{kib}\,\dd\eta$ is totally symmetric with $(M_a)_{ib}=(\Theta_ba)_i$ for $\Theta_b:=\int\partial_bH\,\dd\eta$ . Then, with $\norm{M_a}^2:=\sum_{i,b}(M_a)_{ib}^2$ , $\norm v^2:=\sum_i\norm{v_i}_{L^2(\eta)}^2$ , and all $\Aop$ -pairings read as quadratic-form values,

\boxed{\; a^\top\mathsf Na=1+\norm{M_a}^2+\norm v^2,\qquad a^\top\mathsf Da=\norm{M_a}^2+\inner v{\Aop v},\qquad a^\top\mathsf Ra=1-\inner v{(\Aop-1)v},\;}

(D10.15)

where $\inner v{\Aop v}:=\sum_i\calE(v_i)$ is the closed-form value (written $\calE(v,v)$ in Lemma 16.1; no membership $v_i\in\Dom(\Aop)$ is asserted) and $T_a:=\inner v{(\Aop-1)v}=\sum_i[\calE(v_i)-\norm{v_i}_2^2]\ge0$ . In particular $\mathsf N-\Id\preceq\mathsf D$ .

(e) (Corollaries C1–C5.)

$\mathsf R\preceq\Id$ , and $a^\top\mathsf Ra=1$ for a unit $a$ if and only if every component of the column fluctuation $Ha-a\cdot\one$ lies in the eigenspace $\ker(\Aop-1)$ (“spectrally pure at the gap”).
For $\rho>0$ , $\beta\in\R$ , the bootstrap $(\mathrm{AB})_{\rho,\beta}$ : $\mathsf R\succeq\rho\mathsf N-\beta\Id$ holds at the map if and only if for every unit $a$
$T_a+\rho\bigl(\norm{M_a}^2+\norm v^2\bigr)\le1+\beta-\rho .$
(D10.16)
Since $T_a\ge0$ , $(\mathrm{AB})_{\rho,\beta}$ implies $Q_{\mathrm{lin}}=\lmax(\mathsf N)\le(1+\beta)/\rho$ .
$\mathsf R\succeq\mathsf D$ holds iff $\norm{M_a}^2+\inner v{(2\Aop-1)v}\le1$ for every unit $a$ ; it implies $\sum_b\Theta_b^2\preceq\Id$ , $\norm v^2\le1$ , and $\lmax(\mathsf N)\le2$ .
For the conclusion $Q_{\mathrm{lin}}\le(1+\beta)/\rho$ it suffices that (D10.16) hold at one top eigendirection $a_*$ of $\mathsf N$ .
Summed over an orthonormal basis, (D10.15) shows that the proved trace inequality $\Tr\mathsf R\ge\Tr\mathsf D$ of part (f) is exactly the basis-averaged form of (D10.16) at $(\rho,\beta)=(\tfrac12,0)$ ; the open content of $(\mathrm{AB})$ is its direction-wise de-averaging.

(f) (Unconditional dimension-dependent retention.) $\mathsf D\succeq0$ , $\Tr(H(A+Q))\ge\Tr\bigl(H^{bc}(\partial_bH)(\partial_cH)\bigr)$ pointwise (the cyclic square), and consequently

\Tr\mathsf R\ge\Tr\mathsf D,\qquad \Tr\mathsf N\le2n,\qquad \Tr\mathsf D\le n,\qquad \mathsf R\succeq\mathsf N-n\,\Id,\qquad Q_{\mathrm{lin}}\le n+1 .

(D10.17)

(g) (Products.) If $\mu=\mu_1\otimes\cdots\otimes\mu_n$ is a finite product of centered, variance-one, one-dimensional laws each of the regular class of Definition D10.1, then all matrices block-diagonalize and

\mathsf R-\mathsf D =\diag_k\int V_k''(\psi_k')\,(\psi_k'')^3\,e^{-\psi_k}\,\dd y_k\;\succeq\;0,

(D10.18)

so $\mathsf R\succeq\mathsf D$ , equivalently $\mathsf R\succeq\tfrac12\mathsf N$ , holds with the sharp constants on every such product.

Remark D10.2 (Weak versus strong column equation)

Lemma 16.1 states the column equation in the weak form: the columns $Ha$ lie in $\Dom(\calE)$ and satisfy (D10.13) for every bounded smooth finite-energy $f$ , with $(A+Q)a\in L^1(\eta)$ . That is exactly what part (c) proves, and it is what every downstream identity of this dossier uses (the resolution of part (d) tests (D10.13) against $f=(Ha)_i$ and $f=u_b$ , both in $\mathfrak B$ ). The strong (operator-domain) form — $(Ha)_i\in\Dom(\Aop)$ with $\Aop(Ha)_i=(Ha)_i-((A+Q)a)_i$ in $L^2(\eta)$ , i.e. the operator identity $(1-\Aop)(Ha)=(A+Q)a$ — is not part of the lemma and is not claimed here. It is a separate open question on the compact-target class: by Remark D10.3 it is equivalent to the additional integrability $\Tr Q\in L^2(\eta)$ , which this dossier does not establish on the whole class. It does hold for the products of part (g), where $\Tr Q$ is bounded (Remark D10.6).

1. Pointwise Monge–Ampère calculus (part (a) and the pointwise half of (c))¶

All identities in this subsection are pointwise on $\R^n$ and use only smoothness and strict convexity of $\psi$ and smoothness of $V$ ; no integration is performed.

The Monge–Ampère equation, i.e. the change of variables in $(\nabla\psi)_\#\eta=\mu$ , reads

\log\det H(y)=-\psi(y)+V(\nabla\psi(y)) .

(D10.19)

Differentiating in $y_k$ , with $\partial_k\log\det H=H^{ij}\psi_{ijk}$ :

H^{ij}\psi_{ijk}=-\psi_k+(V_i\circ\nabla\psi)H_{ik} . \tag{MA1}

(D10.20)

Proof D10.1

$\partial_iH^{ij}=-H^{ia}(\partial_iH_{ab})H^{bj}=-H^{bj}\,(H^{ia}\psi_{iab})$ . By (D10.20) with $k=b$ , $H^{ia}\psi_{iab}=-\psi_b+(V_i\circ\nabla\psi)H_{ib}$ , so $\partial_iH^{ij}=H^{bj}\psi_b-(V_i\circ\nabla\psi)H_{ib}H^{bj}=\psi_bH^{bj}-(V_j\circ\nabla\psi)$ . Then

e^{\psi}\partial_i(e^{-\psi}H^{ij}f_j) =H^{ij}f_{ij}+(\partial_iH^{ij})f_j-\psi_iH^{ij}f_j =H^{ij}f_{ij}-(V_j\circ\nabla\psi)f_j .

(D10.22)

Proof D10.2

Chain rule: $f_i=g_a(\nabla\psi)\psi_{ai}=(H\nabla g)_i$ by symmetry of $H$ , which is the first identity; the third follows because $\tau(\nabla\psi(y))=H(y)$ , so $\tau\nabla g\circ\nabla\psi=H\cdot H^{-1}\nabla f=\nabla f$ ; the fourth is then immediate. For the second: $f_{ij}=g_{ab}\psi_{ai}\psi_{bj}+g_a\psi_{aij}$ , so $H^{ij}f_{ij}=g_{ab}H_{ab}+g_a\,H^{ij}\psi_{aij} =\Tr\bigl((D^2g)\,H\bigr)+g_a\bigl(-\psi_a+(V_i\circ\nabla\psi)H_{ia}\bigr)$ by (D10.20), using $H^{ij}\psi_{ai}\psi_{bj}=(HH^{-1}H)_{ab}=H_{ab}$ . Subtracting $(V_i\circ\nabla\psi)f_i=(V_i\circ\nabla\psi)g_aH_{ai}$ leaves $\calL f=\bigl[\Tr(\tau D^2g)-x\cdot\nabla g\bigr]\circ\nabla\psi=(L_\mu g)\circ\nabla\psi$ , since $x=\nabla\psi(y)$ and $g_a\psi_a=(x\cdot\nabla g)\circ\nabla\psi$ .

Lemma D10.2 proves part (a): $U$ intertwines $\calL$ with $L_\mu$ on smooth functions, hence intertwines the closed form $\calE(f)=\int\inner{H^{-1}\nabla f}{\nabla f}\dd\eta$ with the certified target form, and the CMH numerator $\E_\mu\abs{\tau\nabla g}^2$ (isotropic $\Sigma=\Id$ ) is $\int\abs{\nabla_yf}^2\dd\eta$ : on this class the CMH inequality for pullback tests reads $\int\abs{\nabla_yf}^2\dd\eta\le C\int(\calL f)^2\dd\eta$ .

Proof D10.3

Differentiate (D10.20) in $y_\ell$ , using $\partial_\ell H^{ij}=-H^{ia}\psi_{ab\ell}H^{bj}$ :

-H^{ia}H^{bj}\psi_{ab\ell}\psi_{ijk}+H^{ij}\psi_{ijk\ell} =-\psi_{k\ell}+(V_{im}\circ\nabla\psi)\psi_{m\ell}H_{ik} +(V_i\circ\nabla\psi)\psi_{ik\ell} .

(D10.24)

Since $\psi_{ijk\ell}=\partial_{ij}H_{k\ell}$ and $\psi_{ik\ell}=\partial_iH_{k\ell}$ , rearranging gives

\calL H_{k\ell} =H^{ia}H^{bj}\psi_{ab\ell}\psi_{ijk}-H_{k\ell} +H_{ik}(V_{im}\circ\nabla\psi)H_{m\ell},

(D10.25)

and the first term equals $\Tr(H^{-1}\partial_kH\,H^{-1}\partial_\ell H)=Q_{k\ell}$ by total symmetry of $\psi_{abc}$ , while the last is $A_{k\ell}$ . Positivity: for $c\in\R^n$ , $c^\top Ac=\inner{(D^2V\circ\nabla\psi)\,Hc}{Hc}\ge0$ by log-concavity of the target ( $D^2V\succeq0$ on $\operatorname{int}P$ , and $\nabla\psi(y)\in\operatorname{int}P$ ), and $c^\top Qc=\Tr\bigl(H^{-1}S_cH^{-1}S_c\bigr) =\norm{H^{-1/2}S_cH^{-1/2}}_{\HS}^2\ge0$ with $S_c=c_k\partial_kH$ symmetric.

2. The cutoff calculus¶

This subsection is the analytic core: it justifies every integration by parts against the non-compactly-supported core, using only the objects of Definition D10.1 and the import (D10.5).

Proof D10.4

Suppose some sublevel set $\{\psi\le t_0\}$ is unbounded. A closed convex unbounded set contains a ray $\{y_0+sv:s\ge0\}$ , $\abs v=1$ ; convexity of $\psi$ with $\sup_s\psi(y_0+sv)\le t_0$ forces the recession slope of $\psi$ in direction $v$ to be $\le0$ , i.e. $s\mapsto\psi(y+sv)$ is nonincreasing for every $y$ . Then by Fubini along lines in direction $v$ , $\int e^{-\psi}\dd y\ge\int_0^\infty e^{-\psi(y+sv)}\dd s\cdot (\text{transverse integration})=\infty$ for any $y$ with $e^{-\psi(y)}>0$ , contradicting $\eta(\R^n)=1$ . Hence all sublevel sets are compact and the (continuous) minimum is attained.

Fix once and for all $m_0>\min\psi+1$ and, for $m\ge m_0$ , a smooth nonincreasing $\chi_m:\R\to[0,1]$ with $\chi_m=1$ on $(-\infty,m]$ , $\chi_m=0$ on $[2m,\infty)$ , and $\abs{\chi_m'}\le2/m$ . Set

\zeta_m=\chi_m(\psi)\in C_c^\infty(\R^n),\qquad S_m=\operatorname{supp}\nabla\zeta_m\subseteq\{m\le\psi\le2m\} .

(D10.26)

Along the dyadic subsequence $m=2^jm_0$ we may and do choose the $\chi_m$ nested, so that $\zeta_m\uparrow1$ pointwise. Note $\abs{\nabla\zeta_m}\le(2/m)\abs{\nabla\psi}\le2R/m$ uniformly, since $\nabla\psi\in P$ .

Lemma D10.5 (Divergence identities)

(i) For $u,\phi\in C^\infty(\R^n)$ with $\phi$ compactly supported, $\int\phi\,\calL u\,\dd\eta=-\int\inner{H^{-1}\nabla\phi}{\nabla u}\,\dd\eta$ . (ii) For $X\in C^\infty(\R)$ with $X'$ vanishing on $[t,\infty)$ for some $t\in\R$ , $\int\calL\bigl(X(\psi)\bigr)\dd\eta=0$ , where $\calL(X(\psi))=X''(\psi)\,\Gamma+X'(\psi)\bigl(n-\inner{\nabla V\circ\nabla\psi}{\nabla\psi}\bigr)$ and $\Gamma:=\inner{H^{-1}\nabla\psi}{\nabla\psi}$ .

Proof D10.6

Apply Lemma D10.5(ii) with $X'=\chi_m$ :

\int\chi_m'(\psi)\,\Gamma\,\dd\eta =-\int\chi_m(\psi)\,\bigl(n-\inner{\nabla V\circ\nabla\psi}{\nabla\psi}\bigr)\dd\eta .

(D10.27)

Since $\abs{\inner{\nabla V\circ\nabla\psi}{\nabla\psi}}\le c_VR$ and $0\le\chi_m\le1$ , and $\chi_m'\le0$ ,

\int\abs{\chi_m'(\psi)}\,\Gamma\,\dd\eta\le n+c_VR .

(D10.28)

Finally $\inner{H^{-1}\nabla\zeta_m}{\nabla\zeta_m}=\chi_m'(\psi)^2\,\Gamma \le\tfrac2m\abs{\chi_m'(\psi)}\,\Gamma$ , and integrate.

Lemma D10.7 (Pointwise dominations)

Let $\mathfrak g:=H^{bc}\,\partial_bH_{ij}\,\partial_cH_{ij}\ge0$ (full contraction; the integrand of $\Tr\mathsf D$ ). Then, pointwise:

(i) for every $c\in\R^n$ , $\inner{H^{-1}\nabla(c^\top Hc)}{\nabla(c^\top Hc)}\le\abs c^4\,\mathfrak g$ ;

(ii) $\sum_b\norm{\partial_bH}_{\HS}^2\le\norm H_\op\,\mathfrak g\le 2R^2\,\mathfrak g$ ; in particular each $\abs{\psi_{aib}}\le(2R^2\,\mathfrak g)^{1/2}$ ;

(iii) $\abs{\sum_{i,j}H_{ij}\,\inner{H^{-1}\nabla\zeta}{\nabla H_{ij}}} \le\norm H_{\HS}\,\inner{H^{-1}\nabla\zeta}{\nabla\zeta}^{1/2}\,\mathfrak g^{1/2}$ for every smooth $\zeta$ .

Proof D10.7

(i) Let $G_b=c^\top(\partial_bH)c$ , so $\nabla(c^\top Hc)=G$ . For any $\xi\in\R^n$ , with $S_\xi=\xi_b\partial_bH$ , $\xi\cdot G=\inner{S_\xi}{cc^\top}_{\HS}\le\norm{S_\xi}_{\HS}\abs c^2$ . Hence

\inner{H^{-1}G}G=\sup_{\xi\ne0}\frac{(\xi\cdot G)^2}{\inner{H\xi}\xi} \le\abs c^4\sup_{\xi\ne0}\frac{\norm{S_\xi}_{\HS}^2}{\inner{H\xi}\xi} =\abs c^4\sup_{\abs\zeta=1}\norm{S_{H^{-1/2}\zeta}}_{\HS}^2 \le\abs c^4\sum_\beta\norm{S_{H^{-1/2}e_\beta}}_{\HS}^2 =\abs c^4\,\mathfrak g,

(D10.29)

because $\sum_\beta\norm{S_{H^{-1/2}e_\beta}}_{\HS}^2 =\sum_{ij}\inner{H^{-1}\nabla H_{ij}}{\nabla H_{ij}}=\mathfrak g$ . (ii) With the positive semidefinite Gram matrix $\Gamma_{bc}=\inner{\partial_bH}{\partial_cH}_{\HS}$ one has $\mathfrak g=\Tr(H^{-1}\Gamma)\ge\lmin(H^{-1})\Tr\Gamma=\norm H_\op^{-1}\sum_b\norm{\partial_bH}_{\HS}^2$ , and (D10.5) bounds $\norm H_\op$ . The entry bound follows since $\psi_{aib}^2\le\norm{\partial_bH}_{\HS}^2$ . (iii) Cauchy–Schwarz in the $H^{-1}$ -inner product for each $(i,j)$ , then Cauchy–Schwarz over the index sum.

Proof D10.8

Set $\mathcal D_m=\int\zeta_m^2\,\mathfrak g\,\dd\eta$ and $\mathcal T_m=\int\zeta_m^2\,\Tr(H(A+Q))\,\dd\eta$ ; both are finite (compact support, smooth integrands) and $\mathcal T_m\ge0$ since $\Tr(H(A+Q))=\Tr\bigl(H^{1/2}(A+Q)H^{1/2}\bigr)\ge0$ pointwise (Lemma D10.3). For each pair $(i,j)$ apply Lemma D10.5(i) with $u=H_{ij}$ , $\phi=\zeta_m^2H_{ij}$ and sum:

\mathcal D_m =-\int\zeta_m^2\,H_{ij}\,\calL H_{ij}\,\dd\eta -2\int\zeta_m\,H_{ij}\,\inner{H^{-1}\nabla\zeta_m}{\nabla H_{ij}}\,\dd\eta .

(D10.31)

By Lemma D10.3, $-H_{ij}\calL H_{ij}=\Tr(H^2)-\Tr(H(A+Q))$ , so

\mathcal D_m+\mathcal T_m =\int\zeta_m^2\,\Tr(H^2)\,\dd\eta+\mathrm{Err}_m, \qquad \abs{\mathrm{Err}_m}\le2\,\norm{H}_{\HS,\infty}\,\kappa_m^{1/2}\,\mathcal D_m^{1/2} \le\eps_m\,\mathcal D_m^{1/2},

(D10.32)

using Lemma D10.7(iii), Cauchy–Schwarz in $\dd\eta$ , and $\zeta_m\le1$ ; here $\norm H_{\HS,\infty}\le\sqrt n\,2R^2$ by (D10.5) and $\eps_m:=2\sqrt n\,2R^2\,\kappa_m^{1/2}\to0$ by Lemma D10.6. Dropping $\mathcal T_m\ge0$ , $\mathcal D_m\le\Tr\mathsf N+\eps_m\mathcal D_m^{1/2}$ , a quadratic inequality in $\mathcal D_m^{1/2}<\infty$ , whence $\mathcal D_m\le\bigl(\eps_m/2+\sqrt{\Tr\mathsf N+\eps_m^2/4}\bigr)^2$ . Along the nested dyadic sequence $\zeta_m^2\uparrow1$ , monotone convergence gives $\int\mathfrak g\,\dd\eta=\lim\mathcal D_m\le\limsup(\cdots)=\Tr\mathsf N$ . Then $\mathrm{Err}_m\to0$ , and passing to the limit in the displayed identity (monotone convergence on all three nonnegative terms) yields (D10.30).

Proof D10.9

Fix a unit $c$ and let $h=c^\top Hc\in[0,2R^2]$ . By Lemma D10.3, $c^\top(A+Q)c=\calL h+h\ge0$ pointwise. By Lemma D10.5(i) with $\phi=\zeta_m$ , $u=h$ :

\int\zeta_m\,c^\top(A+Q)c\,\dd\eta =\int\zeta_m h\,\dd\eta-\int\inner{H^{-1}\nabla\zeta_m}{\nabla h}\,\dd\eta .

(D10.33)

The error term is bounded by $\kappa_m^{1/2}\bigl(\int_{S_m}\inner{H^{-1}\nabla h}{\nabla h}\dd\eta\bigr)^{1/2} \le\kappa_m^{1/2}\bigl(\int\mathfrak g\,\dd\eta\bigr)^{1/2}\to0$ by Lemma D10.7(i) and Lemma D10.8. Since $\zeta_m\uparrow1$ and the integrand is nonnegative, monotone convergence on the left and dominated convergence on $\int\zeta_mh\,\dd\eta\to\int h\,\dd\eta=c^\top\Id c=1$ (using $\int H\,\dd\eta=\E_\mu\tau=\Id$ , the certified isotropic normalization) give $\int c^\top(A+Q)c\,\dd\eta=1=\abs c^2$ . Polarizing over $c$ (all quantities finite) gives $\int(A+Q)\dd\eta=\Id$ and, taking $c=e_k$ and summing, $\Tr(A+Q)\in L^1$ ; entries are dominated by the trace of a positive semidefinite matrix.

Lemma D10.10 (Form membership and weak pairings)

Let $\mathfrak B$ be the set of $f\in C^\infty(\R^n)$ (source coordinates) that are bounded with $\calE^0(f):=\int\inner{H^{-1}\nabla f}{\nabla f}\dd\eta<\infty$ . Then:

(i) every $f\in\mathfrak B$ lies in $\Dom(\calE)$ (the closed form domain) with $\calE(f)=\calE^0(f)$ , and $\calE(f,g)=\int\inner{H^{-1}\nabla f}{\nabla g}\dd\eta$ for $f,g\in\mathfrak B$ ;

(ii) every column component $H_{ai}$ , every $u_b$ , and every core pullback belongs to $\mathfrak B$ ;

(iii) for every $f\in\mathfrak B$ and every unit $a$ , the weak column equation (D10.13) holds, all pairings being absolutely convergent.

Proof D10.10

(i) Let $f\in\mathfrak B$ and let $g=f\circ(\nabla\psi)^{-1}$ , smooth and bounded on $\operatorname{int}P$ . The target-side cutoff $\zeta_m\circ(\nabla\psi)^{-1}$ is smooth with support $\nabla\psi(\{\psi\le2m\})$ , a compact subset of $\operatorname{int}P$ (Lemma D10.4 and the diffeomorphism property); hence $(\zeta_m\circ(\nabla\psi)^{-1})\,g$ extends by zero to an element of $C_c^\infty(\R^n)$ and $\zeta_mf$ corresponds to a core element of $\mathscr D$ . Moreover

\calE^0(\zeta_mf-f) \le2\int(1-\zeta_m)^2\inner{H^{-1}\nabla f}{\nabla f}\dd\eta +2\norm f_\infty^2\,\kappa_m\longrightarrow0

(D10.34)

by dominated convergence and Lemma D10.6, while $\zeta_mf\to f$ in $L^2(\eta)$ . Closedness of $\calE$ gives $f\in\Dom(\calE)$ with $\calE(f)=\lim\calE^0(\zeta_mf) =\calE^0(f)$ ; the bilinear statement follows by polarization ( $\mathfrak B$ is a vector space). (ii) $H_{ai}$ is smooth, bounded by (D10.5), and $\calE^0(H_{ai})\le\int\mathfrak g\,\dd\eta<\infty$ (Lemma D10.8); $u_b=\partial_b\psi$ is smooth, bounded by $R$ , with $\calE^0(u_b)=\int H^{cd}H_{bc}H_{bd}\dd\eta=\int H_{bb}\dd\eta=1$ ; a core pullback $f=c+g\circ\nabla\psi$ , $g\in C_c^\infty$ , is smooth and bounded with $\nabla f=H\nabla g$ bounded, hence $\calE^0(f)=\int\inner{H\nabla g}{\nabla g}\dd\eta\le\norm{\nabla g}_\infty^2\,n<\infty$ . (iii) Fix $f\in\mathfrak B$ and write $w_i=(Ha)_i=a_kH_{ki}$ , a linear combination of entries, so $w_i\in\mathfrak B$ . Lemma D10.5(i) with $\phi=\zeta_mf$ , $u=w_i$ :

\int\zeta_mf\,\calL w_i\,\dd\eta =-\int\zeta_m\inner{H^{-1}\nabla f}{\nabla w_i}\dd\eta -\int f\inner{H^{-1}\nabla\zeta_m}{\nabla w_i}\dd\eta .

(D10.35)

As $m\to\infty$ : the left side converges to $\int f\,\calL w_i\,\dd\eta$ by dominated convergence, since $\calL w_i=\bigl((A+Q)a\bigr)_i-w_i$ (Lemma D10.3) is dominated by $\Tr(A+Q)+2R^2\in L^1$ (Lemma D10.9, positivity of $A+Q$ ) and $f$ is bounded; the first right-hand term converges to $-\int\inner{H^{-1}\nabla f}{\nabla w_i}\dd\eta=-\calE(f,w_i)$ by dominated convergence (the integrand is dominated by the product of two $L^2$ functions); the second is bounded in absolute value by $\norm f_\infty\,\kappa_m^{1/2}\,\calE^0(w_i)^{1/2}\to0$ . Rearranging gives (D10.13).

3. Eigenstructure and the Brascamp–Lieb gap (part (b))¶

Proof D10.11

$u_b=x_b\circ\nabla\psi$ with $x_b$ the target coordinate, so $\int u_b\,\dd\eta=\E_\mu X_b=0$ (centering) and $\int u_bu_c\,\dd\eta=\E_\mu[X_bX_c]=\delta_{bc}$ (isotropy) — no integration by parts is needed. Work in target coordinates. Choose $\chi\in C_c^\infty(\R^n)$ with $\chi=1$ on a neighborhood of $\overline P$ ; then $\ell_b:=(x_b\chi)|_{\operatorname{int}P}=x_b$ on the support and $\ell_b\in\mathscr D$ is a core element. For any core $f=c+F|_{\operatorname{int}P}$ , $F\in C_c^\infty(\R^n)$ , the certified weak Stein identity of Theorem 4.1 (in the ambient test form $\int x_iF\,\dd\mu=\int\tau_{ij}\partial_jF\,\dd\mu$ , exactly as used by the certified recovery dossier) gives

\calE^0(f,\ell_b) =\E_\mu\inner{\tau\nabla f}{e_b} =\int\tau_{bj}\,\partial_jF\,\dd\mu =\int x_b\,F\,\dd\mu =\int x_b\,f\,\dd\mu ,

(D10.36)

the last step because $\E_\mu X_b=0$ kills the constant. Both sides are continuous for the form norm of $f$ (Cauchy–Schwarz for the form; $x_b$ bounded for the pairing), and the core is dense in $\Dom(\calE)$ by construction of the closure, so $\calE(f,\ell_b)=\inner f{\ell_b}_{L^2(\mu)}$ for all $f\in\Dom(\calE)$ . By definition of the self-adjoint operator of a closed form, $\ell_b\in\Dom(\Aop)$ and $\Aop\ell_b=\ell_b$ . Transporting by $U$ gives the source statement; consistently, $\calL u_b=-u_b$ pointwise is exactly (D10.20).

Proof D10.12

The Brascamp–Lieb inequality Brascamp & Lieb, 1976 for the log-concave probability $\eta=e^{-\psi}\dd y$ with $D^2\psi=H\succ0$ states $\Var_\eta(f)\le\int\inner{H^{-1}\nabla f}{\nabla f}\dd\eta$ for every $C^1$ (or locally Lipschitz) $f$ for which the right side is finite; this applies to every core pullback (smooth, bounded, bounded gradient). For general $f\in\Dom(\calE)$ take core $f_k\to f$ in form norm: then $\Var(f_k)\to\Var(f)$ ( $L^2$ -convergence of functions and of their means) and $\calE(f_k)\to\calE(f)$ , so the inequality passes to the closure. For centered $f$ , $\Var(f)=\norm f_2^2$ , i.e. $\inner f{\Aop f}\ge\norm f_2^2$ in the form sense on $\one^\perp\cap\Dom(\calE)$ , which is the spectral statement. The value 1 is attained because $\Aop u_b=u_b$ with $u_b\perp\one$ .

4. The column equation and its compatibilities (part (c))¶

Parts of (c) already proved: positivity and the pointwise identity (Lemma D10.3), $L^1$ -normalization (Lemma D10.9), form membership and the weak column equation (D10.13) (Lemma D10.10). It remains to record orthogonality and the third-moment identity.

Proof D10.14

Finiteness: $\abs{\psi_{kib}}\le(2R^2\mathfrak g)^{1/2}\in L^2(\eta)\subset L^1(\eta)$ by Lemma D10.7(ii) and Lemma D10.8; symmetry is the symmetry of third derivatives. For the identity, integrate by parts with the cutoffs: $u_be^{-\psi}=\psi_be^{-\psi}=-\partial_b(e^{-\psi})$ , so for each $m$ , by the divergence theorem for the compactly supported field $\zeta_mH_{ki}e^{-\psi}e_b$ ,

\int\zeta_m\,H_{ki}\,u_b\,\dd\eta =\int\zeta_m\,\psi_{kib}\,\dd\eta+\int H_{ki}\,\partial_b\zeta_m\,\dd\eta .

(D10.38)

The last term is bounded by $2R^2\cdot(2R/m)\to0$ (uniform gradient bound on $\zeta_m$ ), and the first two converge by dominated convergence ( $H_{ki}u_b$ bounded; $\psi_{kib}\in L^1$ ).

Remark D10.3 (The precise open gap: strong column equation)

For fixed $a$ , the following are equivalent by the definition of the operator of a closed form: (a) $(Ha)_i\in\Dom(\Aop)$ for all $i$ with $\Aop(Ha)=(Ha)-(A+Q)a$ in $L^2(\eta;\R^n)$ ; (b) $(A+Q)a\in L^2(\eta;\R^n)$ . Since $\abs{(A+Q)a}\le\Tr(A+Q)$ pointwise and $\Tr A\le n\,c_{V''}(2R^2)^2$ is bounded, (b) holds whenever $\Tr Q\in L^2(\eta)$ . This dossier proves $\Tr Q\in L^1(\eta)$ (Lemma D10.9) but not $\Tr Q\in L^2(\eta)$ on the general compact-target class. Lemma 16.1 asserts only the weak form (D10.13), which is proved (Lemma D10.10); the strong form (a) is therefore not a claim of the lemma, and whether it holds on the whole class is a separate open question, equivalent to $\Tr Q\in L^2(\eta)$ . Every statement of Theorem D10.1 downstream uses only the weak form. On the products of part (g) the strong form does hold (Remark D10.6).

5. Proof of the resolution (part (d))¶

Fix a unit $a$ and write $w_i=(Ha)_i$ . By Lemma D10.10, $w_i\in\Dom(\calE)$ ; by the certified normalization $\int H\,\dd\eta=\Id$ , $\inner{w_i}{\one}=a_i$ . Define

v_i:=w_i-a_i\one-\sum_b(M_a)_{ib}\,u_b\;\in\;\Dom(\calE),

(D10.39)

so that $v_i\perp\one$ and $v_i\perp u_b$ for all $b$ by construction (Lemma D10.14 and $\int u_b\dd\eta=0$ ), and $v_i\in\mathfrak B$ (a finite linear combination of $\mathfrak B$ -elements). This is the decomposition $Ha=a\cdot\one+\sum_b(M_a)_{\cdot b}u_b+v$ of the statement, with $U=\operatorname{span}\{u_1,\dots,u_n\}$ ; no claim is made that $U$ exhausts $\ker(\Aop-1)$ , and no operator inverse is used.

First identity. Pointwise $\sum_iw_i^2=(H^2)_{aa}$ , so $a^\top\mathsf Na=\sum_i\norm{w_i}_2^2$ . The three parts of $w_i$ are pairwise orthogonal in $L^2(\eta)$ ( $\one\perp u_b$ by centering; $\one,u_b\perp v_i$ by construction; $u_b\perp u_c$ , $b\ne c$ , by isotropy), so Pythagoras gives $\norm{w_i}_2^2=a_i^2+\sum_b(M_a)_{ib}^2+\norm{v_i}_2^2$ ; summing over $i$ with $\sum_ia_i^2=1$ :

a^\top\mathsf Na=1+\norm{M_a}^2+\norm v^2 .

(D10.40)

Second identity. By definition of $\mathsf D$ and Lemma D10.10(i),

a^\top\mathsf Da =\sum_i\int\inner{H^{-1}\nabla w_i}{\nabla w_i}\dd\eta =\sum_i\calE(w_i).

(D10.41)

Expand $\calE(w_i)$ bilinearly. $\calE(\one,\cdot)=0$ . For the $u$ -modes, the form–operator pairing with $\Aop u_b=u_b$ gives $\calE(u_b,u_c)=\inner{u_b}{u_c}=\delta_{bc}$ and $\calE(u_b,v_i)=\inner{u_b}{v_i}=0$ . Hence $\calE(w_i)=\sum_b(M_a)_{ib}^2+\calE(v_i)$ , and summing over $i$ :

a^\top\mathsf Da=\norm{M_a}^2+\inner v{\Aop v}, \qquad \inner v{\Aop v}:=\sum_i\calE(v_i)<\infty .

(D10.42)

Third identity. Subtract, using the manuscript definition $\mathsf R=\mathsf N-\mathsf D$ :

a^\top\mathsf Ra =1+\norm v^2-\inner v{\Aop v} =1-\sum_i\bigl[\calE(v_i)-\norm{v_i}_2^2\bigr] =1-T_a,

(D10.43)

and $T_a\ge0$ because each $v_i$ is centered and Lemma D10.12 applies. This proves (D10.15). Finally $a^\top(\mathsf N-\Id-\mathsf D)a=\norm v^2-\inner v{\Aop v}\le0$ , i.e.\ $\mathsf N-\Id\preceq\mathsf D$ (the componentwise Brascamp–Lieb inequality of the probe, here a one-line consequence of the resolution). $\square$

6. Corollaries C1–C5 (part (e))¶

C1. $a^\top\mathsf Ra=1-T_a\le1$ for every unit $a$ , so $\mathsf R\preceq\Id$ . Equality in direction $a$ means $T_a=0$ , i.e. $\calE(v_i)=\norm{v_i}_2^2$ for every $i$ . Each $v_i$ is centered, so its spectral measure under $\Aop$ is carried by $[1,\infty)$ (Lemma D10.12), and $\calE(v_i)-\norm{v_i}_2^2=\int_{[1,\infty)}(\lambda-1)\,\dd\inner{E_\lambda v_i}{v_i}=0$ iff the spectral measure of $v_i$ is concentrated at $\{1\}$ , iff $v_i\in\ker(\Aop-1)$ (possibly $v_i=0$ ). Since each $u_b\in\ker(\Aop-1)$ , this holds iff every component of the column fluctuation $Ha-a\cdot\one=\sum_b(M_a)_{\cdot b}u_b+v$ lies in $\ker(\Aop-1)$ : spectral purity at the gap. Note the criterion refers to the full eigenspace $\ker(\Aop-1)$ , of which $U$ may be a proper subspace; the dossier nowhere needs them to coincide.

C2. By the resolution, for a unit $a$ ,

a^\top\mathsf Ra-\rho\,a^\top\mathsf Na+\beta =1-T_a-\rho\bigl(1+\norm{M_a}^2+\norm v^2\bigr)+\beta,

(D10.44)

which is $\ge0$ iff (D10.16) holds. $(\mathrm{AB})_{\rho,\beta}$ is the conjunction over all unit $a$ . If it holds, then since $T_a\ge0$ , $\rho\,a^\top\mathsf Na\le1+\beta-T_a\le1+\beta$ , so $Q_{\mathrm{lin}}=\lmax(\mathsf N)\le(1+\beta)/\rho$ ; this recovers the w3c01 bootstrap conclusion without using $\mathsf N-\Id\preceq\mathsf D$ .

C3. $a^\top(\mathsf R-\mathsf D)a =1-T_a-\norm{M_a}^2-\inner v{\Aop v} =1-\norm{M_a}^2-\inner v{(2\Aop-1)v}$ , using $T_a+\inner v{\Aop v}=2\inner v{\Aop v}-\norm v^2$ . So $\mathsf R\succeq\mathsf D$ iff $\norm{M_a}^2+\inner v{(2\Aop-1)v}\le1$ for all unit $a$ . Since $\inner v{(2\Aop-1)v}\ge\norm v^2\ge0$ (Lemma D10.12), this implies $\norm{M_a}^2\le1$ and $\norm v^2\le1$ for all $a$ , hence $a^\top\mathsf Na\le2$ . Finally $\norm{M_a}^2=\sum_{i,b}(\Theta_ba)_i^2=a^\top\bigl(\sum_b\Theta_b^2\bigr)a$ by Lemma D10.14 ( $\Theta_b$ symmetric), so $\norm{M_a}^2\le1$ for all unit $a$ is exactly $\sum_b\Theta_b^2\preceq\Id$ . Also, through $\mathsf N=\mathsf D+\mathsf R$ (Proposition D10.1 or the definition of $\mathsf R$ ), $\mathsf R\succeq\mathsf D\iff2\mathsf R\succeq\mathsf N\iff \mathsf R\succeq\tfrac12\mathsf N$ .

C4. Let $a_*$ be a unit eigenvector of $\mathsf N$ at $\lmax(\mathsf N)$ . If (D10.16) holds at $a_*$ , then as in C2, $\rho\lmax(\mathsf N)=\rho\,a_*^\top\mathsf Na_*\le1+\beta$ . Only the top direction is used.

C5. Summing (D10.15) over an orthonormal basis $a=e_1,\dots,e_n$ : $\Tr\mathsf N=n+\sum_a(\norm{M_{e_a}}^2+\norm{v^{(e_a)}}^2)$ and $\Tr\mathsf R=n-\sum_aT_{e_a}$ . Hence the inequality $\sum_aT_{e_a}+\tfrac12\sum_a(\norm{M_{e_a}}^2+\norm{v^{(e_a)}}^2)\le\tfrac n2$ is literally equivalent to $\Tr\mathsf R\ge\tfrac12\Tr\mathsf N$ , i.e. (via $\Tr\mathsf N=\Tr\mathsf D+\Tr\mathsf R$ ) to $\Tr\mathsf R\ge\Tr\mathsf D$ — which part (f) proves. Thus the Chen–Klartag-type trace inequality is exactly the orthonormal-basis average of the channel inequality (D10.16) at $(\rho,\beta)=(\tfrac12,0)$ , and the open content of $(\mathrm{AB})$ is its direction-wise de-averaging. $\square$

7. Unconditional dimension-dependent retention (part (f))¶

Lemma D10.15 (Pointwise cyclic square)

Pointwise on $\R^n$ ,

\Tr(HQ)-\Tr\bigl(H^{bc}(\partial_bH)(\partial_cH)\bigr) =\frac16\sum_{p,q,r}\frac{\widetilde T_{pqr}^2}{\lambda_p\lambda_q\lambda_r} \Bigl[(\lambda_p-\lambda_q)^2+(\lambda_q-\lambda_r)^2+(\lambda_r-\lambda_p)^2\Bigr]\ge0,

(D10.45)

where, at the given point, $H=\sum_p\lambda_pe_pe_p^\top$ is a spectral decomposition and $\widetilde T_{pqr}=e_p^ie_q^je_r^k\,\psi_{ijk}$ . Moreover $\Tr(HA)=\Tr(H^3(D^2V\circ\nabla\psi))\ge0$ .

Proof D10.16

Both sides are scalars; expand in the spectral decomposition. With $\widetilde T$ totally symmetric,

\Tr(HQ)=\sum_{p,q,r}\frac{\lambda_r}{\lambda_p\lambda_q}\,\widetilde T_{pqr}^2, \qquad \Tr\bigl(H^{bc}(\partial_bH)(\partial_cH)\bigr) =\sum_{p,q,r}\frac1{\lambda_p}\,\widetilde T_{pqr}^2 .

(D10.46)

(For the first: $Q_{k\ell}=\sum_{p,q}(\lambda_p\lambda_q)^{-1} (\partial_kH)_{ij}e_p^ie_q^j(\partial_\ell H)_{i'j'}e_p^{i'}e_q^{j'}$ and contract $k,\ell$ against $H=\sum_r\lambda_re_re_r^\top$ ; for the second contract $b,c$ against $H^{-1}=\sum_r\lambda_r^{-1}e_re_r^\top$ and the free matrix indices against $\Id=\sum_pe_pe_p^\top$ .) Since $\widetilde T_{pqr}^2$ is symmetric under permutations of $(p,q,r)$ , replace each coefficient by its symmetrization:

\frac13\Bigl[\frac{\lambda_r}{\lambda_p\lambda_q}+\frac{\lambda_p}{\lambda_q\lambda_r} +\frac{\lambda_q}{\lambda_r\lambda_p}\Bigr] -\frac13\Bigl[\frac1{\lambda_p}+\frac1{\lambda_q}+\frac1{\lambda_r}\Bigr] =\frac{\lambda_p^2+\lambda_q^2+\lambda_r^2-\lambda_p\lambda_q-\lambda_q\lambda_r -\lambda_r\lambda_p}{3\lambda_p\lambda_q\lambda_r},

(D10.47)

and the numerator is $\tfrac12[(\lambda_p-\lambda_q)^2+(\lambda_q-\lambda_r)^2 +(\lambda_r-\lambda_p)^2]\ge0$ . For the last claim, $\Tr(HA)=\Tr(H\cdot H(D^2V\circ\nabla\psi)H)=\Tr\bigl(H^{3/2}(D^2V\circ\nabla\psi)H^{3/2}\bigr)\ge0$ by cyclicity and $D^2V\succeq0$ , $H^{3/2}\succ0$ .

Proof D10.17 (Proof of part (f))

$\mathsf D\succeq0$ : for $c\in\R^n$ , $c^\top\mathsf Dc=\int H^{bc'}\inner{(\partial_bH)c}{(\partial_{c'}H)c}\dd\eta\ge0$ (Gram sum against the positive matrix $H^{-1}$ ); alternatively it is a sum of form values. By Lemma D10.15, pointwise $\Tr(H(A+Q))\ge\Tr(HQ)\ge\mathfrak g$ , and all three are nonnegative and integrable (Lemma D10.8), so by (D10.30)

\Tr\mathsf R=\Tr\mathsf N-\Tr\mathsf D=\int\Tr(H(A+Q))\,\dd\eta\ \ge\ \int\mathfrak g\,\dd\eta =\Tr\mathsf D .

(D10.48)

Hence $\Tr\mathsf N\ge2\Tr\mathsf D$ . Tracing $\mathsf N-\Id\preceq\mathsf D$ (part (d)) gives $\Tr\mathsf N-n\le\Tr\mathsf D\le\tfrac12\Tr\mathsf N$ , whence $\Tr\mathsf N\le2n$ and $\Tr\mathsf D\le n$ . Then for every unit $a$ , $a^\top\mathsf Ra=a^\top\mathsf Na-a^\top\mathsf Da\ge a^\top\mathsf Na-\Tr\mathsf D \ge a^\top\mathsf Na-n$ , i.e. $\mathsf R\succeq\mathsf N-n\Id$ ( $a^\top\mathsf Da\le\Tr\mathsf D$ because $\mathsf D\succeq0$ ). Finally $a^\top\mathsf Na\le1+a^\top\mathsf Da\le1+n$ , so $Q_{\mathrm{lin}}=\lmax(\mathsf N)\le n+1$ .

8. Products of one-dimensional laws (part (g))¶

Proof D10.18 (Proof of part (g))

Let $\mu=\bigotimes_{k=1}^n\mu_k$ with each $\mu_k=e^{-V_k}\one_{(\alpha_k,\beta_k)}\dd x_k$ centered, of variance 1, $V_k\in C^\infty(\R)$ convex on $(\alpha_k,\beta_k)$ ; then $\mu$ satisfies Definition D10.1 with $P=\prod_k[\alpha_k,\beta_k]$ . Let $\psi_k$ be the canonical moment potential of $\mu_k$ . The sum $\psi(y)=\sum_k\psi_k(y_k)$ is smooth, strictly convex, satisfies the Monge–Ampère equation (D10.19) of the product (both sides factorize), and pushes $e^{-\psi}\dd y=\bigotimes_ke^{-\psi_k}\dd y_k$ (a probability) forward to $\mu$ ; by the essential uniqueness of the moment potential up to source translation Cordero-Erausquin & Klartag, 2015, and because every matrix in (D10.9) is invariant under source translations (it is an integral of a function of derivatives of $\psi$ against $e^{-\psi}$ ), we may compute with this $\psi$ .

Now $H=\diag(\psi_k''(y_k))$ and $\psi_{ijk}=0$ unless $i=j=k$ , so all objects are diagonal:

A=\diag\bigl(V_k''(\psi_k')\,(\psi_k'')^2\bigr),\qquad Q=\diag\Bigl(\frac{(\psi_k''')^2}{(\psi_k'')^2}\Bigr),\qquad H^{bc}(\partial_bH)(\partial_cH)=\diag\Bigl(\frac{(\psi_k''')^2}{\psi_k''}\Bigr),

(D10.49)

and $\mathsf N,\mathsf D,\mathsf R$ are diagonal with $k$ -th entries equal to the corresponding one-dimensional integrals (Fubini; each integrand depends on $y_k$ alone). In one dimension the pointwise identity

H\,Q=\psi_k''\cdot\frac{(\psi_k''')^2}{(\psi_k'')^2} =\frac{(\psi_k''')^2}{\psi_k''}=\bigl(H^{bc}(\partial_bH)(\partial_cH)\bigr)_{kk}

(D10.50)

holds exactly (no inequality). By Proposition D10.1, applied to the product map (which is in the class, so the proposition is available), the $k$ -th diagonal entry of $\mathsf R$ is $\int H(A+Q)_{kk}\dd\eta=\int\bigl(V_k''(\psi_k')(\psi_k'')^3+(\psi_k''')^2/\psi_k''\bigr) e^{-\psi_k}\dd y_k$ , whence

(\mathsf R-\mathsf D)_{kk} =\int V_k''(\psi_k')\,(\psi_k'')^3\,e^{-\psi_k}\,\dd y_k\;\ge\;0

(D10.51)

by convexity of $V_k$ . Diagonal matrices compare entrywise in the Loewner order, so $\mathsf R\succeq\mathsf D$ , and $\mathsf N=\mathsf D+\mathsf R$ gives $\mathsf R\succeq\tfrac12\mathsf N$ .

9. Calibrations (remarks only; no proof step depends on them)¶

Remark D10.7 (Exact boundary calibrations)

The following one-dimensional laws are boundary calibrations: the Gaussian has noncompact target, and the one-sided exponential, symmetric Laplace, and exponential products have nonsmooth or noncompactly-supported densities, so none of them belongs to the regular class of Definition D10.1. They are limits of regular laws and are recorded only to display the exact values of the resolution channels; each value below is an elementary closed-form integral against the classical Stein kernels ( $\tau=1$ Gaussian; $\tau(x)=x+1$ for the centered one-sided exponential; $\tau(x)=\abs x/\sqrt2+\tfrac12$ for the isotropic symmetric Laplace).

\begin{array}{l|ccccc} \text{law} & \mathsf N & \mathsf D & \mathsf R & T_a & \norm{M_a}^2\\\hline \text{Gaussian (any }n) & \Id & 0 & \Id & 0 & 0\\ \text{one-sided exponential (1D)} & 2 & 1 & 1 & 0 & 1\\ \text{symmetric Laplace (1D)} & 5/4 & 1/2 & 3/4 & 1/4 & 0\\ \text{product of one-sided exponentials} & 2\Id & \Id & \Id & 0 & 1\ (\text{all unit }a) \end{array}

(D10.52)

Three structural observations. (i) The exponential saturates both $\mathsf R\preceq\Id$ and $\mathsf R\succeq\mathsf D$ with zero high-mode excess: its column fluctuation $\tau-1=x$ is exactly the gap eigenfunction, so the extremal configuration of the linear gate is spectrally pure at the Brascamp–Lieb gap; for the product, the totally symmetric tensor $M_{kib}=\delta_{kib}$ gives $\norm{M_a}^2=\abs a^2=1$ for every unit $a$ . (ii) The Laplace shows $T_a>0$ occurs: the excess channel is real, not vacuous. (iii) All rows satisfy $\mathsf N=\mathsf D+\mathsf R$ and the resolution identities, provided the $A$ -term is read distributionally where $V$ is nonsmooth — see the next warning.

Distributional curvature of the Laplace

The isotropic symmetric Laplace has $V''=2\sqrt2\,\delta_0$ as a measure. The retention $\mathsf R-\mathsf D=\E_\mu[V''\tau^3]$ of the one-dimensional identity is carried entirely by this atom:

\E_\mu[V''\tau^3] =2\sqrt2\;\tau(0)^3\rho(0) =2\sqrt2\cdot\tfrac18\cdot\tfrac1{\sqrt2}=\tfrac14,

(D10.53)

which is exactly what reconciles $\mathsf R=3/4$ with $\mathsf D=1/2$ . Smooth regular approximants spread this atom over a shrinking interval. Any computation on a piecewise-smooth target that discards the $A$ -term where $V$ fails to be twice differentiable loses this mass and produces wrong values; this is a concrete audit warning for future work on boundary laws.

10. Audit trail¶

Hypotheses actually used. (1) Definition D10.1 (smooth positive density on a convex body, centered, isotropic, log-concave target), through the imported Theorem 4.1 (regularity, diffeomorphism, weak Stein identity with zero flux, $\E_\mu\tau=\Id$ ). (2) The published pointwise bound (D10.5) Klartag, 2014, used in: finiteness of $\mathsf N$ ; boundedness of $u_b,H_{ai},A$ ; Lemma D10.7–Lemma D10.10; Remark D10.6. (3) The classical Brascamp–Lieb inequality Brascamp & Lieb, 1976 (Lemma D10.12 only). (4) Essential uniqueness of the moment potential Cordero-Erausquin & Klartag, 2015 (part (g) only). (5) Definition 16.1 operator conventions (form core, closability, self-adjoint $\Aop$ ); prop:cmh-bochner is a listed ledger dependency (depends_on of lem:cmh-linear-spectral-resolution) but no identity of this dossier consumes it, so under CLAUDE.md constraint 8 it should be dropped from depends_on when the proofs[] record is wired; the repair handoff proposes that delta. No other external input is used; in particular no unreviewed preprint import is load-bearing (the trace bound $\Tr\mathsf N\le2n$ is re-derived, not imported).

Flagged gaps. None. The statement of Lemma 16.1 now agrees with Theorem D10.1: the column equation is stated and proved in the weak form (D10.13) (columns in $\Dom(\calE)$ , $(A+Q)a\in L^1(\eta)$ , bounded smooth finite-energy test functions), and $\inner v{\Aop v}$ is the closed-form value. The strong operator-domain form $(1-\Aop)(Ha)=(A+Q)a$ is not asserted by the lemma; it is equivalent to $\Tr Q\in L^2(\eta)$ , open on the general class (Remark D10.3) and proved on products (Remark D10.6). No step is conditional: given the published imports above, Theorem D10.1 as stated is unconditional on the class of Definition D10.1.

Obstructions respected. The ledger node carries no bounded_by edge; the six obstruction statements of the manuscript are scoped to the Eldan fixed-cut program, and the route guardrails are checked one by one. rem:two-tail-slice-bounds: no localization cut, slice bound, or covariance-weighted estimate appears; all statements are stationary identities at one fixed map. rem:projection-ceiling: the controlled quantities are full column energies $\int\abs{Ha}^2\dd\eta$ and full slice norms $\norm{M_a}$ ; no scalar projection bound is promoted. rem:crude-insufficient and rem:relative-ceiling: no stochastic covariance integral, bootstrap, or all-measure relative bound occurs; part (f) is explicitly dimension-dependent. rem:profile-circularity: no isoperimetric profile or evolving competitor family occurs; Brascamp–Lieb is a certified external input, not an assumed profile bound. rem:single-coordinate-cuts: products enter only through exact stationary block-diagonalization. CMH guardrails: no pointwise Loewner promotion of the cyclic square is asserted (Lemma D10.15 is used under the trace only; the w3c01 jet fence is respected); the arguments consume differentiated Monge–Ampère structure throughout, as prop:letwin-not-gate-zero requires of any statement of this strength (the eigen-equation for $u_b$ , the column equation, and (D10.30) all come from (D10.20)–Lemma D10.3); no canonical kernel is transported through a noninvertible map; no continuity or semicontinuity of $Q_{\mathrm{lin}}$ or $\CMH$ is asserted; boundary laws appear only as calibration remarks; the trace-upgrade cluster is not opened and no comparison with it is made.

References¶

Berman, R. J., & Berndtsson, B. (2013). Real Monge–Ampère Equations and Kähler–Ricci Solitons on Toric Log Fano Varieties. Annales de La Faculté Des Sciences de Toulouse. Mathématiques, 22(4), 649–711. 10.5802/afst.1386
Fathi, M. (2019). Stein Kernels and Moment Maps. The Annals of Probability, 47(4), 2172–2185. 10.1214/18-AOP1305
Klartag, B. (2014). Logarithmically-Concave Moment Measures I. In Geometric Aspects of Functional Analysis (Vol. 2116, pp. 231–260). Springer. 10.1007/978-3-319-09477-9_16
Brascamp, H. J., & Lieb, E. H. (1976). On Extensions of the Brunn–Minkowski and Prékopa–Leindler Theorems, Including Inequalities for Log Concave Functions, and with an Application to the Diffusion Equation. Journal of Functional Analysis, 22(4), 366–389. 10.1016/0022-1236(76)90004-5
Cordero-Erausquin, D., & Klartag, B. (2015). Moment Measures. Journal of Functional Analysis, 268(12), 3834–3866. 10.1016/j.jfa.2015.04.001